Page 491 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

τὴν πυραμίδα, ἧς βάσις μὲν τὸ ΘΟΕΠΖΡΗΣ πολύγωνον, to circle EFGH [Prop. 12.2], thus as circle ABCD (is)
κορυφὴ δὲ τὸ Ν σημεῖον. καὶ ὡς ἄρα ὁ ΑΛ κῶνος πρὸς τὸ to circle EFGH, so polygon DT AUBV CW also (is) to
Ξ στερεόν, οὕτως ἡ πυραμίς, ἧς βάσις μὲν τὸ ΔΤΑΥΒΦΓΧ polygon HPEQFRGS. And as circle ABCD (is) to cir-
πολύγωνον, κορυφὴ δὲ τὸ Λ σημεῖον, πρὸς τὴν πυραμίδα, cle EFGH, so cone AL (is) to solid O. And as poly-
ἧς βάσις μὲν τὸ ΘΟΕΠΖΡΗΣ πολύγωνον, κορυφὴ δὲ τὸ Ν gon DT AUBV CW (is) to polygon HPEQFRGS, so the
σημεῖον· ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ ΑΛ κῶνος πρὸς τὴν ἐν pyramid whose base is polygon DT AUBV CW, and apex
αὐτῷ πυραμίδα, οὕτως τὸ Ξ στερεὸν πρὸς τὴν ἐν τῷ ΕΝ the point L, (is) to the pyramid whose base is polygon
κώνῳ πυραμίδα. μείζων δὲ ὁ ΑΛ κῶνος τῆς ἐν αὐτῷ πυ- HPEQFRGS, and apex the point N [Prop. 12.6]. And,
ραμίδος· μεῖζον ἄρα καὶ τὸ Ξ στερεὸν τῆς ἐν τῷ ΕΝ κώνῳ thus, as cone AL (is) to solid O, so the pyramid whose
πυραμίδος. ἀλλὰ καὶ ἔλασσον· ὅπερ ἄτοπον. οὐκ ἄρα ἐστὶν base is DT AUBV CW, and apex the point L, (is) to the
ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως ὁ ΑΛ pyramid whose base is polygon HPEQFRGS, and apex
κῶνος πρὸς ἔλασσόν τι τοῦ ΕΝ κώνου στερεόν. ὁμοίως the point N [Prop. 5.11]. Thus, alternately, as cone AL
δὲ δείξομεν, ὅτι οὐδέ ἐστιν ὡς ὁ ΕΖΗΘ κύκλος πρὸς τὸν is to the pyramid within it, so solid O (is) to the pyramid
ΑΒΓΔ κύκλον, οὕτως ὁ ΕΝ κῶνος πρὸς ἔλασσόν τι τοῦ within cone EN [Prop. 5.16]. But, cone AL (is) greater
ΑΛ κώνου στερεόν. than the pyramid within it. Thus, solid O (is) also greater
Λέγω δή, ὅτι οὐδέ ἐστιν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν than the pyramid within cone EN [Prop. 5.14]. But, (it
ΕΖΗΘ κύκλον, οὕτως ὁ ΑΛ κῶνος πρὸς μεῖζόν τι τοῦ ΕΝ is) also less. The very thing (is) absurd. Thus, circle
κώνου στερεόν. ABCD is not to circle EFGH, as cone AL (is) to some
Εἰ γὰρ δυνατόν, ἕστω πρὸς μεῖζον τὸ Ξ· ἀνάπαλιν ἄρα solid less than cone EN. So, similarly, we can show that
ἐστὶν ὡς ὁ ΕΖΗΘ κύκλος πρὸς τὸν ΑΒΓΔ κύκλον, οὕτως neither is circle EFGH to circle ABCD, as cone EN (is)
τὸ Ξ στερεὸν πρὸς τὸν ΑΛ κῶνον. ἀλλ᾿ ὡς τὸ Ξ στερεὸν to some solid less than cone AL.
πρὸς τὸν ΑΛ κῶνον, οὕτως ὁ ΕΝ κῶνος πρὸς ἔλασσόν τι So, I say that neither is circle ABCD to circle EFGH,
τοῦ ΑΛ κώνου στερεόν· καὶ ὡς ἄρα ὁ ΕΖΗΘ κύκλος πρὸς as cone AL (is) to some solid greater than cone EN.
τὸν ΑΒΓΔ κύκλον, οὕτως ὁ ΕΝ κῶνος πρὸς ἔλασσόν τι For, if possible, let it be (in this ratio) to (some)
τοῦ ΑΛ κώνου στερεόν· ὅπερ ἀδύνατον ἐδείχθη. οὐκ ἄρα greater (solid), O. Thus, inversely, as circle EFGH is to
ἐστὶν ὡς ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως ὁ circle ABCD, so solid O (is) to cone AL [Prop. 5.7 corr.].
ΑΛ κῶνος πρὸς μεῖζόν τι τοῦ ΕΝ κώνου στερεόν. ἐδείχθη But, as solid O (is) to cone AL, so cone EN (is) to some
δέ, ὅτι οὐδὲ πρὸς ἔλασσον· ἔστιν ἄρα ὡς ὁ ΑΒΓΔ κύκλος solid less than cone AL [Prop. 12.2 lem.]. And, thus, as
πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως ὁ ΑΛ κῶνος πρὸς τὸν ΕΝ circle EFGH (is) to circle ABCD, so cone EN (is) to
κῶνον. some solid less than cone AL. The very thing was shown
᾿Αλλ᾿ ὡς ὁ κῶνος πρὸς τὸν κῶνον, ὁ κύλινδρος πρὸς (to be) impossible. Thus, circle ABCD is not to circle
τὸν κύλινδρον· τριπλασίων γὰρ ἑκάτερος ἑκατέρου. καὶ ὡς EFGH, as cone AL (is) to some solid greater than cone
ἄρα ὁ ΑΒΓΔ κύκλος πρὸς τὸν ΕΖΗΘ κύκλον, οὕτως οἱ ἐπ᾿ EN. And, it was shown that neither (is it in this ratio) to
αὐτῶν ἰσοϋψεῖς. (some) lesser (solid). Thus, as circle ABCD is to circle
Οἱ ἄρα ὑπὸ τὸ αὐτὸ ὕψος ὄντες κῶνοι καὶ κύλινδροι EFGH, so cone AL (is) to cone EN.
ibþ EFGH, as (the ratio of the cylinders) on them (having)
But, as the cone (is) to the cone, (so) the cylin-
πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις· ὅπερ ἔδει δεῖξαι. der (is) to the cylinder. For each (is) three times each
[Prop. 12.10]. Thus, circle ABCD (is) also to circle

the same height.
Thus, cones and cylinders having the same height are
to one another as their bases. (Which is) the very thing it
was required to show.

Proposition 12
.
Οἱ ὅμοιοι κῶνοι καὶ κύλινδροι πρὸς ἀλλήλους ἐν τρι- Similar cones and cylinders are to one another in the
πλασίονι λόγῳ εἰσὶ τῶν ἐν ταῖς βάσεσι διαμέτρων. cubed ratio of the diameters of their bases.
῎Εστωσαν ὅμοιοι κῶνοι καὶ κύλινδροι, ὧν βάσεις μὲν Let there be similar cones and cylinders of which the
οἱ ΑΒΓΔ, ΕΖΗΘ κύκλοι, διάμετροι δὲ τῶν βάσεων αἱ ΒΔ, bases (are) the circles ABCD and EFGH, the diameters
ΖΘ, ἄξονες δὲ τῶν κώνων καὶ κυλίνδρων οἱ ΚΛ, ΜΝ· λέγω, of the bases (are) BD and FH, and the axes of the cones

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